We obtain an exact expression for the conductivity of a non-Galilean-invariant Fermi liquid with impurities by solving the kinetic equation with both screened Coulomb and $z=3$ Pomeranchuk critical interactions. While consistent with previous asymptotic results, our solution shows that electron-electron interactions enter the conductivity solely via the quasiparticle scattering time, ${\tau }_{\mathrm{ee}}$. Accordingly, the crossovers between the collisionless and hydrodynamic regimes occur when $1/{\tau }_{\mathrm{ee}}$ becomes comparable to the larger of the impurity scattering rate and the probe frequency, $\mathrm{\Omega }$. Importantly, this crossover is governed by the quasiparticle scattering rate, rather than by a suppressed effective current-relaxation rate. The exact solution also shows that the electron-electron correction to the $\mathrm{dc}$ conductivity remains parametrically smaller than the residual conductivity throughout the Fermi-liquid regime. For the optical conductivity, our solution also provides access to the hydrodynamic regime, $\mathrm{\Omega }\ll 1/{\tau }_{\mathrm{ee}}$, which is inaccessible within perturbation theory. Near a $z=3$ Pomeranchuk quantum critical point, consistency between the kinetic equation and Kubo approaches requires proper inclusion of mass renormalization within the Eliashberg approximation, which also ensures that the crossover between the collisionless and hydrodynamic regimes in the optical conductivity occurs at the Planckian scale $\mathrm{\Omega }\sim T$.

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• Electrical conductivity
• Optical conductivity
• Fermi liquid theory